Demonstrating the Evolution of Complex Genetic Representations: … · 2017-10-30 · Algorithms (EDAs, see Pelikan, Goldberg, & Lobo 1999 for a review). These algorithms structure

Demonstrating the Evolution of

Complex Genetic Representations:

An Evolution of Artificial Plants

Marc Toussaint

Institut fur Neuroinformatik, Ruhr-Universitat Bochum, ND 04, 44780 Bochum—Germany

[email protected]

April 5, 2003

In Genetic and Evolutionary Computation Conference (GECCO 2003), 2003. In Press.

Abstract. A common idea is that complex evolutionary adaptation is enabled by complex genetic

representations of phenotypic traits. This paper demonstrates how, according to a recently developed

theory, genetic representations can self-adapt in favor of evolvability, i.e., the chance of adaptive

mutations. The key for the adaptability of genetic representations is neutrality inherent in non-trivial

genotype-phenotype mappings and neutral mutations that allow for transitions between genetic

representations of the same phenotype. We model an evolution of artificial plants, encoded by

grammar-like genotypes, to demonstrate this theory.

1 Introduction

In ordinary evolutionary systems, including natural evolution and standard Genetic Algorithms(GAs), the search strategy is determined by uncorrelated mutations and recombinations on thelevel of genes. In view of this trial-and-error search strategy, it might seem surprising that naturalevolution was that effective in finding highly structured, complex solutions. One might have ex-pected that the search strategy needs to be more sophisticated, more structured and adapted tothe “objective function” to accomplish this efficiency.

An example for sophisticated, adapted, and structured search strategies are recent developmentsin evolutionary computation, which can be collected under the name of Estimation-of-Distribution

2 1 INTRODUCTION

Algorithms (EDAs, see Pelikan, Goldberg, & Lobo 1999 for a review). These algorithms structuresearch by learning probabilistic models of the distribution of good solutions.

Now, it hardly seems that ordinary evolutionary systems like natural evolution or GAs learn“models about the distribution of good solutions” and adapt their mutational exploration accord-ingly. However, it is possible for ordinary evolutionary systems to adapt their search strategy andnatural evolution certainly exploits this possibility. The key to understand this possibility is toconsider a non-trivial relation between genotype and phenotype, between the genes and the phenes(phenotypic traits that are fitness-relevant). In this case it is possible that the same phenotypecan be represented by different genotypes. In other words, evolution could principally adapt theway it represents solutions.

Adapting the genetic representation of phenes while keeping mutation operators fixed is in somesense dual to adapting the mutation operators while keeping the representation of phenes fixed.Especially in the biology literature it is often argued that this is the way natural evolution adapts itssearch strategy: The way genes encode phenotypic traits is surely not an incident but the outcomeof a long adaptative process in favor of evolvability (Wagner & Altenberg 1996). Much research isspend to understand these principles of gene interactions (like epistasis or canalization) (Altenberg1995; Wagner, Booth, & Bagheri-Chaichian 1997; Wagner, Laubichler, & Bagheri-Chaichian 1998;Hansen & Wagner 2001a; Hansen & Wagner 2001b).

But there would be another open question. Do genetic representations really adapt such thatthe search strategy becomes “better”? Is the induced adaptation of the search strategy similarto the adaptation in the case of EDAs, such that indeed a model of the distribution of goodsolutions is indirectly learned by adapting the representations of solutions? Recently, a theory onthe adaptation of phenotypic search distributions (exploration distributions) in the case of non-trivial genotype-phenotype mappings was proposed (Toussaint 2003). Since the present paper aimsto illustrate this process and thereby to open a more intuitive access to this theory, we want, atthis place, briefly review the main results.

A non-trivial genotype-phenotype mapping is such that (1) the same phenotype is encoded bymore than one genotype, and (2), at least for some of these phenotypes, the induced phenotypicvariability depends on the genotype it is encoded with. In this case, every genotype g may bewritten as a tuple (x, σ), where x is its phenotype and σ generally represents any kinds of neutraltraits of that genotype (strategy parameters are only a special case). Different σ for the same xmean different genetic representations of the same phenotype. In (Toussaint 2003) it is shown thata σ is one-to-one identifiable with a mutation distribution (i.e., search distribution) “around” g.Hence, the tuple (x, σ) actually lives in the product space of phenotype space and the space ofdistributions.

Now, the main approach is the following: Instead of investigating the evolution of genotypes gin a given evolutionary process, one can also analyze how x and σ evolve in that same process.The most interesting result is that the equation that describes the evolution of σ’s turns out tobe itself an equation similar to evolutionary dynamics. The equation allows to associate a qualitymeasure to each σ and says that σ’s evolve as to increase this quality measure just as normalevolutionary processes evolve as to increase the fitness of genotypes. This quality measure of σ’scan be interpreted as a measure of “how good the mutational phenotypic variability (the phenotypicsearch distribution) matches the distribution of good organisms”—just as for EDAs. The resultsare summarized in following corollary:

Corollary 1.1 (Toussaint 2003). The evolution of genetic representations, or equivalently, ofexploration distributions (σ-evolution) naturally has a selection pressure towards

Marc Toussaint—April 5, 2003 3

– minimizing the Kullback-Leibler divergence between exploration and the exponential fitnessdistribution, and

– minimizing the entropy of exploration.

Here, the exponential fitness distribution describes the distribution of good solutions and theKullback-Leibler divergence is a distance measure between two distributions and thus measures thematch between the search strategy (as given by the exploration distribution) and this exponentialfitness distribution.

This paper will present a first case study for the evolution of genetic representations by modelingan evolution of artificial plants. We chose this case study for several reasons: First, the genotype-phenotype mapping that we will introduce shortly is highly non-trivial in the strict sense we definedit for σ-evolution, and it is interpretable from an (abstract) biological point of view. Further, thiskind of encoding and the possibility to visually display the phenotypes allows to really graspthe genetic encodings, how completely different genetic representations of the same phenotype arepossible, that complex features like gene interactions and modularity of the genetic representationscan emerge.

The next section introduces the genetic encoding and the mutability we assume on this encod-ing. The major novelty here are 2nd-type mutations that allow for neutral transitions betweenequivalent genetic representations within the grammar-like encoding. Section 3 then explains theexperimental setup and presents the experiments that demonstrate the evolution of complex geneticrepresentations. Conclusions follow thereafter.

2 A Genotype Model and 2nd-type Mutations

The genotype-phenotype mapping we will assume is a variant of the L-systems proposed byPrusinkiewicz & Hanan (1989) to encode plant-like structures. Let us introduce this encodingfrom another point of view that puts more emphasis on the principles of ontogenesis and geneinteractions.

Let us consider the development of an organism as an interaction of its state Ψ (describing,e.g., its cellular configuration) and a genetic system Π, neglecting environmental interactions.Development begins with an initial organism in state Ψ(0), the “egg cell”, which is also inheritedand which we model as part of the genotype. Then, by interaction with the genetic system, theorganism develops through time, Ψ(1),Ψ(2), .. ∈ P , where P is the space of all possible organismstates. Hence, the genetic system may be formalized as an operator Π : P → P modifyingorganism states such that Ψ(t) = Πt Ψ(0). We make a specific assumption about this operator: weassume that Π comprises a whole sequence of operators, Π = 〈π1, π2, .., πr〉, each πi : P → P . Asingle operator πi (also called production rule) is meant to represent a transcription module, i.e.,a single gene or an operon. Based on these ideas we define the general concept of a genotype anda genotype-phenotype mapping for our model:

A genotype consists of an initial organism Ψ(0) ∈ P and a sequence Π = 〈π1, π2, .., πr〉, πi : P →P of operators. A genotype-phenotype mapping φ develops the final phenotype Ψ(T ) by recursivelyapplying all operators on Ψ(0).

This definition is somewhat incomplete because it does not explain the stopping time T ofdevelopment and in which order operators are applied. We keep to the simplest options: we applythe operators in sequential order and will fix T to some chosen value.

4 2 A GENOTYPE MODEL AND 2ND-TYPE MUTATIONS

For the experiments we need to define how we represent an organism state Ψ and how operatorsare applied. We represent an organism by a sequence of symbols 〈ψ1, ψ2, ..〉, ψi ∈ A. Each symbolmay be interpreted, e.g., as the state of a cell; we choose the sequence representation as thesimplest spatially organized assembly of such states. Operators are represented as replacementrules 〈a0:a1, a2, ..〉, ai ∈ A, that apply on the organism by replacing all occurrences of a symbol a0

by the sequence 〈a1, a2, ..〉. If the sequence 〈a1, a2, ..〉 has length greater than 1, the organism isgrowing; if it has length 0, the organism shrinks. Calling a0 promoter and 〈a1, a2, ..〉 the structuralgenes gives the analogy to operons in natural genetic systems. For example, if the initial organismis given by Ψ(0) = 〈a〉 and the genetic system is Π =

⟨〈a:ab〉, 〈a:cd〉, 〈b:adc〉

⟩, then the organism

grows as: Ψ(0) = 〈a〉, Ψ(1) = 〈cdadc〉, Ψ(2) = 〈cdcdadcdc〉, etc.

The general idea is that these operators are basic mechanisms which introduce correlatingeffects between phenotypic traits. Riedl (1977) already claimed that the essence of the operonis to introduce correlations between formerly independent genes in order to adopt the functionaldependence between the genes and their phenotypic effects and thereby increase the probability ofsuccessful variations.

The proposed model is very similar to the models by Kitano (1990), Gruau (1995), and Lucas(1995), who use grammar-encodings to represent neural networks. It is also comparable to newapproaches to evolve complex structures by means of so-called symbiotic composition (Hornby &Pollack 2001a; Hornby & Pollack 2001b; Watson & Pollack 2002).

The crucial novelty in our model are 2nd-type mutations . These allow for genetic variationsthat explore the neutral sets which are typical for any grammar-like encoding. Without theseneutral variations, self-adaptation of genetic representations and of exploration distributions is notpossible.

Consider the three genotypes given in the first column,genotype phenotype phenotypic neighbors

Ψ(0) = 〈a〉, Π =⟨〈a:bcbc〉

⟩〈bcbc〉 〈*〉, 〈*cbc〉, 〈b*bc〉, 〈bc*c〉, 〈bcb*〉

Ψ(0) = 〈a〉, Π =⟨〈a:dd〉, 〈d:bc〉

⟩〈bcbc〉 〈*〉, 〈*bc〉, 〈bc*〉, 〈*c*c〉, 〈b*b*〉

Ψ(0) = 〈bcbc〉, Π =⟨ ⟩

〈bcbc〉 〈*cbc〉, 〈b*bc〉, 〈bc*c〉, 〈bcb*〉All three genotypes have, after developing for at least two time steps, the same phenotype Ψ(t) =〈bcbc〉, t ≥ 2. The third genotype resembles what one would call a direct encoding, where thephenotype is directly inherited as Ψ(0). Assume that, during mutation, all symbols, except forthe promoters, mutate with fixed, small probability. By considering all one-point mutations of thethree genotypes, we get the phenotypic neighbors of 〈bcbc〉 as given in the third column of the ta-ble, where a star * indicates the mutated random symbol. This shows that, although all the threegenotypes represent the same phenotype, they induce completely different phenotypic variabili-ties, i.e., completely different phenotypic search distributions. Note that, wherever a phenotypicneighbor comprises two stars, these two phenotypic variations are completely correlated.

In order to enable a variability of genetic representations within such a neutral set we need toallow for mutational transitions between phenotypically equivalent genotypes. A transition fromthe 1st to the 3rd genotype requires a genetic mutation that applies the operator 〈a:bcbc〉 on theegg cell 〈a〉 and deletes it thereafter. Both, the application of an operator on some sequence (beit the egg cell or another operator’s) and the deletion of operators will be mutations we providein our model. The transition from the 2nd to the 1st genotype is similar: the 2nd operator 〈d:ab〉is applied on the sequence of the first operator 〈a:dd〉 and deleted thereafter. But we must alsoaccount for the inverse of these transitions. A transition from the 3rd genotype to the 1st is possible

Marc Toussaint—April 5, 2003 5

• First type mutations are ordinary symbol mutations that occur in every sequence (i.e., promoter or rhs

of an operator) of the genotype; namely symbol replacement, symbol duplication, and symbol deletion,

which occur with equal probabilities. The mutation frequency for every sequence is Poisson distributed

with the mean number of mutations given by (α · sequence-length), where α is a global mutation rate

parameter.

• The second type mutations aim at a neutral restructuring of the genetic system. A 2nd-type mutation

occurs by randomly choosing an operator π and a sequence p from the genome, followed by one of the

following operations:

– application of an operator π on a sequence p,

– inverse application of an operator π on a sequence p; this means that all matches of the operator’s

rhs sequence with subsequences of p are replaced in p by the operator’s promoter,

– deletion of an operator π, only if it was never applied during ontogenesis,

– application of an operator π on all sequences of the genotype followed by deletion of this operator,

– generation of a new operator ν by extracting a random subsequence of stochastic length 2 +

Poisson(1) from a sequence p and encoding it in an operator with random promoter. The new

operator ν is inserted in the genome behind the sequence p, followed by the inverse application

of ν on p.

All these mutations occur with equal probabilities. The total number of second type mutations for a

genotype is Poisson distributed with mean β. The second type mutations are not necessarily neutral

but they are neutral with sufficient probability to enable an exploration of neutral sets.

• A genotype is mutated by first applying second type mutations and thereafter first type mutations,

the frequencies of which are given by β and α, respectively.

Table 1: The mutation operators in our model.

if a new operator is created by randomly extracting a subsequence (here bcbc from the egg cell)and encoding it in a new operator (here 〈a:bcbc〉). The original subsequence is then replaced by thepromoter. Similarly, a transition from the 1st to the 2nd genotype occurs when the subsequencebc is extracted from the operator 〈a:bcbc〉 and encoded in a new operator 〈d:bc〉.

Basically, the mutations we will provide in our model are the generation of new operators byextraction of subsequences (deflation), and the application and deletion of existing operators (infla-tion). Technical details can be found in table 1; the main point of these mutation operators thoughis not their details but that they in principle enable a transition between phenotypically equivalentrepresentations in our encoding. We call these mutations 2nd-type mutations to distinguish themfrom ordinary symbol mutation, deletion, and insertion.

3 Evolving Plants

The symbols for encoding plants. The sequences we are evolving are strings of the alphabet{A,..,P} which are mapped on plant-describing strings according to {A,..,I} 7→ {F,+,-,&,^,\,/,[,]}and {J,..,P} 7→ {.}. The meanings of the symbols of the plant-describing strings are summarizedin table 2. For example, the sequence 〈FF[+F][-F]〉 represents a plant for which the stem growstwo units upward before it branches in two arms, one to the right, the other to the left, each ofwhich has one unit length and a leave attached at the end.

Table 3 demonstrates the implications of this encoding. The plant on the left is an examplestaken from (Prusinkiewicz & Lindenmayer 1990). To the right of this “original” you find three

6 3 EVOLVING PLANTS

F attach a unit length stem

+,-,&

^,\,/

rotations of the “local coordinate system” (that apply on following attached units): four rotation

of δ degree in all four directions away from the stem (+,-,&,^) and two rotations of ±δ degree

around the axis of the stem (\,/).

[ a branching (instantiation of a new local coordinate system)

] the end of that branch: attaches a leave and proceeds with the previous branch (return to the

previous local coordinate system); also if the very last symbol of the total sequence is not ],

another leave is attached

. does nothing

Table 2: Description of the plant grammars symbols, cf. (Prusinkiewicz & Hanan 1989).

〈F:FF-[-F+F+F]+[+F-F-F]〉 〈F:FF-[+F+F+F]+[+F-F-F]〉 〈F:FF-[-F-F+F]+[+F-F-F]〉

Table 3: An example for a 2D plant structure and its phenotypic variability induced by single symbol

mutations in the genotype. For all of them T = 4, δ = 20, and Ψ(0)=〈F〉. Below each illustration, the

single production rule that was mutated is given.

different variations, each of which was produced by a single symbol mutation in the genotype.Obviously, there are large-scale correlations in the phenotypic variability induced by uncorrelatedgenetic mutations.

The fitness function. Given a sequence we evaluate its fitness by first drawing the corresponding3D plant in a virtual environment (the OpenGL 3D graphics environment). We chop off everythingof the plant that is outside a bounding cube of size b×b×b. Then we grab a bird’s view perspective ofthis plant and measure the area of green leaves as observed from this perspective. The measurementis also height dependent: the higher a leave (measured by OpenGL’s depth buffer in logarithmicscale, where 0 corresponds to the cube’s floor and 1 to the cube’s ceiling), the more it contributesto the green area integral

L =∫

x∈bird’s view area

[color of x = green

]·[height at x

] dAreab2

∈ [0, 1] . (1)

This integral is the positive component of a plant’s fitness. The negative component is related tothe number and “weight” of branch elements: To each element i we associate a weight wi which isdefined recursively. The weight of a leave is 1; the total weight of a subtree is the sum of weights ofall the elements of that subtree; and the weight of a branch element i is 1 plus the total weight ofthe subtree that is attached to this branch element. E.g., a branch that has a single leave attachedhas weight 1 + 1 = 2, a branch that has two branches each with a single leave attached has weight1 + (2 + 1) + (2 + 1) = 7, etc. The idea is that wi roughly reflects how “thick” i has to be in order

Marc Toussaint—April 5, 2003 7

1st trial 2nd trial

δ, b 20,20 ←↩ L-system angle δ, size of the bounding cube

T 1 ←↩ stopping time of development

% 10−7 10−6 factor of the weight term W in a plant’s fitness f

α .01 ←↩ (initial) frequency of first type mutations

β .01 .3 (initial) frequency of second type mutations

τ (τ ′) .5 (.1) 0 (0) rate of self-adaptation of α and β

µ, λ ←↩ type of selection

no ←↩ crossover turned on?

〈AAAFFFJ〉 ←↩ initialization of Ψ(0) of all genotypes in the first popula-

tion (they have no operators)

A {A,..,P} ←↩ symbol alphabet

λ 100 ←↩ (offspring) population size

µ 30 ←↩ (selected) parent population size

Mmax 100 000 1 000 000 maximal number of symbols in a phenotype

Rmax 100 ←↩ maximal number of operators allowed in one genotype

Umax 40 ←↩ symbol duplication mutations are allowed only if a se-

quence has less than Umax symbols

Table 4: Parameter settings of the two trials. The “←↩” means “same as for the previous trial” and shows

that only few parameters are changed from trial to trial.

to carry the attached weight. The total weight of the whole tree,

W =∑

i

wi ,

gives the negative component of a plant’s fitness. In our experiments we used

f = L − %W

as the fitness function, where the penalty factor % was chosen % ∈ {10−6, 10−7} in the differentexperiments.

Details of the implementation. Evolving such plant structures already gets close to the limits oftoday’s computers, both, with respect to memory and computation time. Hence, we use some ad-ditional techniques to improve efficiency: First, we impose different limits on the memory resourcethat a genotype and phenotype is allowed to allocate, namely three: (1) The number of symbols ina phenotype was limited to be lower or equal than a number Mmax. This holds in particular duringontogenesis: If the application of an operator results in a phenotype with too many symbols, thenthe operator is simply skipped. (2) The number of operators in a genotype is limited to be ≤Rmax. If a mutation operator would lead to more chromosomes, this mutation operator is simplyskipped (no other mutation is made in place). (3) There is a soft limit on the number of symbolsin a single chromosome: A duplication mutation is skipped if the chromosome already has length≥ Umax. The limit though does not effect 2nd-type mutations; an inflative mutation π ·p may verywell lead to chromosomes of length greater than Umax.

Second, we adopt an elaborated technique of self-adaptation of the mutation frequencies. Weused the scheme similar to the self-adaptation of strategy parameters proposed by Back (1996).

8 3 EVOLVING PLANTS

0 2000 4000 6000 80000

0.2

0.4

0.6

0.8

1

1.2

1st t

rial

fitness#elements /10000genome size /1000*operator usage /5

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

1.2

2nd

trial

fitness#elements /10000genome size /1000*operator usage /10

Figure 1: The graphs display the curves of the fitness, the number of phenotypic elements, the genome size,

and the operators usage of the best individual in all generations of the trials. Note that every quantity has

been rescaled to fit in the range of the ordinate, e.g., the number of phenotype elements has been divided

by 10 000 as indicated by the notation “#elements/10000”. The * for the operator usages indicates that

the curve has been smoothed by calculating the running average over an interval of about a hundredth of

the abscissae (80 and 20 in the respective graphs).

Every genome i additionally encodes two real numbers αi and βi. Before any other mutations aremade, they are mutated by

αi ← αi (S N(0, τ) + τ ′) , (2)

βi ← βi (S N(0, τ) + τ ′) ,

where S N(0, τ) is a random sample (independently drawn for αi and βi) from the Gaussian dis-tribution N(0, τ) with zero mean and standard deviation τ . The parameter τ ′ allows to induce apressure towards increasing mutation rates. After this mutation, αi and βi determine the mutationfrequencies of 1st- and 2nd-type mutations respectively.

The 1st trial. Let us discuss two of the trials made with different parameters. Table 4 summarizesthe experimental setup. See figure 1. For the 1st trial, the curves show some sudden changes atgeneration ∼4000 where the fitness, the number of phenotypic elements, the number of operatorsin the genomes, and the total genome length explodes. Between generation ∼4000 and ∼5400, themost significant curve in the graph is the repeatedly decaying genome size. Indeed we will find thatthe genomes in this period are too large and mutationally unstable. The innovations extinct andgenome size decays until at generation ∼5400 a comparably large number of phenotypic elementscan be encoded by much smaller genomes that comprise more operators. In table 5, the illustrationsof the best individual in selected generations explain in more detail what happened. For a longtime this is not much until, in generation 4000, a couple of leaves turn up at certain places of thephenotype. Then, very rapidly, more leaves pop up until, in generation 4025, every phenotypicsegment has a leave attached. This is exactly what we call a correlated phenotypic adaptation andwas enabled by encoding all the segments that now carry a leave within one operator, namely theA-operator. The resulting “long-arm-building-block” triggers a revolution in phenotypic variability

Marc Toussaint—April 5, 2003 9

3800: f=.0034 e=49 w=612 o=2 g=66/2

Π=〈〈N:NNAA〉,〈M:MMMPNNAA〉〉4025: f=.0087 e=156 w=1813 o=1 g=114/1

Π=〈〈A:IMAJA〉〉4400: f=.28 e=3467 w=92031 o=5 g=512/5

Π=〈〈K:KA〉,〈J:64〉,〈B:BF〉,〈G:GIFJCIJAIJA〉,〈O:29〉〉

5100: f=.20 e=1479 w=57134 o=2 g=217/2

Π=〈〈K:〉,〈J:77〉〉5400: f=.22 e=1410 w=59219 o=3 g=159/3

Π=〈〈I:AJ〉,〈C:JJCJ〉,〈J:64〉〉8000: f=.31 e=3652 w=379288 o=2 g=141/2

Π=〈〈I:39〉,〈J:60〉〉

Table 5: The 1st trial. The illustrations display the phenotypes at selected generations. The two squared

pictures in the lower right corner of each illustration display exactly the bird’s view perspective that is used

to calculate the fitness: The lower (colored) picture displays the plant as seen from above and determines

which area enters the green area integral in equation (1), and the upper gray-scale picture displays the

height value of each element which enters the same equation (where white and black refer to height 0 and

1, respectively). Below each illustration you find some data corresponding to this phenotype: 〈generation〉:f=〈fitness〉 e=〈number of elements〉 w=〈plant’s total weight〉 o=〈number of used operators〉 g=〈genome

size〉/〈number of operators in the genome〉. The genetic system Π is also displayed. (Ψ(0) is generally too

large to be displayed here.) For some operators, the size of the rhs is given instead of the sequence. See

the text for a discussion of this evolution.

and leads to the large structures as illustrated for generation 4400 (3467 elements). However, thesestructures are not encoded efficiently, the genome size is too large (512) and phenotypic variabilitybecomes chaotic. The species almost extinguishes until, in generation 5100, evolution finds a muchbetter structured genome to encode large phenotypes. The J-operator becomes dominant andallows to encode 1479 phenotypic elements with a genome size of 217. This concept is furtherimproved and evolves until, in generation 8000, a genome of size 141 with 2 operators encodes aregularly structured phenotype of 3652 elements.

The 2nd trial. For the 2nd trial we turned off the self-adaptation mechanism for the mutationfrequencies (based on the experience with previous trials we can now estimate a good choice ofα = .01 and β = .3 for the mutation frequencies and fix it) and increase the limit Mmax tomaximally 1 000 000 elements per phenotype. The severe change in the resulting structures is alsodue to the increase of the weight penalty factor % to 10−6—the final structure of the 1st trial

10 3 EVOLVING PLANTS

950: f=.018 e=144 w=2681 o=3 g=163/3

Π=〈〈N:IP〉,〈P:JF〉,〈F:IIAJJF〉〉1010: f=.032 e=506 w=10250 o=2 g=211/2

Π=〈〈N:NIJFFFFFFFFF〉,〈F:IIAAJJF〉〉

1650: f=.052 e=1434 w=31476 o=5 g=180/8

Π=〈〈B:IJNN〉,〈N:IAAJFIAAJF〉,〈N:4〉,〈B:3〉,〈B:9〉,〈F:IINFNNKK

KCBJF〉,〈B:IJNNAJ〉,〈N:IJBA〉〉

1900: f=.17 e=4915 w=105099 o=10 g=230/12

Π=〈〈B:NN〉,〈N:IABFIAAJF〉,〈N:

33〉,〈J:MOFJ〉,〈D:KNME〉,〈B:BB

BBJ〉,〈B:FB〉,〈B:MNNLDDM〉,〈F:

32〉,〈B:28〉,〈N:NLA〉,〈L:IJ〉〉

1910: f=.20 e=4340 w=89996 o=12 g=226/14

Π=〈〈B:NN〉,〈N:IABFIAAJF〉,〈N:

33〉,〈J:MMJ〉,〈J:NJ〉,〈D:KNME〉,〈B:BBBBJ〉,〈B:FB〉,〈B:MNNLD

DM〉,〈F:30〉,〈B:18〉,〈C:KCKKC

LK〉,〈N:NLA〉,〈L:IJ〉〉

2100: f=.33 e=9483 w=192235 o=10 g=261/15

Π=〈〈B:NN〉,〈N:IAABFIAAJF〉,〈N:57〉,〈J:B〉,〈J:JJ〉,〈D:DOIE〉,〈B:FBFBFBFBHJ〉,〈B:ENNDD〉,〈F:36〉,〈B:25〉,〈B:6〉,〈C:CCGLB〉,〈B:CLK〉,〈N:NNLA〉,〈L:IJ〉〉

Table 6: The 2nd trial. Please see the caption of table 5 for explanations.

has a weight of about .3 · 10−6 which would now lead to a crucial penalty. The weight punishingfactor % enforces structures that are regularly branched instead of long curling arms. Table 6presents the results of the 2nd trial. Comparing the illustrations for generation 950 and 1010 wesee that evolution very quickly developed a fan-like structure that is attached at various placesof the phenotype. The fans arise from an interplay of two operators: The N-operator encodesthe fan-like structures while the F-operator encodes the spokes of these fans. Adaptation of thesefans is a beautiful example for correlated exploration. The N-operator encodes more and morespokes until the fan is complete in generation 1010, while the F-operator makes the spokes longer.Elongation proceeds and results in the “hairy”, long-armed structures. Note that, in generation1650, one N- and two B-operators are redundant. Until generation 1900, leaves are attached toeach segment of the arms, similar to generation 4025 of the 1st trial. At that time, the plant’sweight is already 105 099 and probably prohibits to make the arms even longer (since weight wouldincrease exponentially). Instead a new concept develops: At the tip of each arm two leaves are nowattached instead of one and this quickly evolves until there are three leaves, in generation 1910,and eventually a complete fan of six leaves attached at the tip of each arm. In generation 2100,a comparably short genome with 10 used operators encodes a very dense phenotype structure of9483 elements.

More trials, data, and source code can be found at the author’s home page.

4 Conclusions

Let us briefly discuss whether similar result could have been produced with a more conventional GAthat uses, instead of our non-trivial genotype-phenotype mapping, a direct encoding of sequencesin {F,+,-,&,^,\,/,[,]} that describe the plants. For example, setting β = 0 in our modelcorresponds to such a GA since no operators will be created and the evolution takes places solelyon the “egg cell” Ψ(0), which is equal to the final phenotype in the absence of operators. We do notneed to present the results of such a trial—not much happens. The obvious reason is the unsolvabledilemma of long sequences in a direct encoding: On the one hand, mutability must be small suchthat long sequences can be represented stably below the error threshold of reproduction; on theother hand mutability should not vanish in order to allow evolutionary progress. This dilemmabecomes predominant when trying to evolve sequences of length ∼104, as it is the case for theplants evolved in the 2nd trial. Also elaborate methods of self-adaptation of the mutation ratecannot circumvent this problem completely; the only way to solve the dilemma is to allow for anadaptation of genetic representations. The key novelty in our model that enabled the adaptationof genetic representations are the 2nd-type mutations we introduced.

In our example, two important features of the genetic representations coincide. First, this isthe capability to find compact representations that allow to encode large phenotypes with smallgenotypes solving the error threshold dilemma. Second, this is the ability for complex adaptation,i.e., to induce highly structured search distributions that incorporate large-scale correlations be-tween phenotypic traits. For example, the variability of one leave is, in certain representations, notindependent of the variability of another leave. A GA with direct encoding would have to optimizeeach single phenotypic element by itself, step by step. The advantage of correlated exploration isthat many phenotypic elements can be adapted simultaneously in dependence of each other.

Our experiments demonstrated the theory of σ-evolution which mainly states that the evolutionof genetic representations is guided by a fundamental principle: they evolve such that the matchbetween the evolutionary search distribution and the distribution of good solutions becomes better.The way genetic systems are organized is a mirror of what evolution has learned about the problem.

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